The L.C.M. of 114 and 95 is:

Options

A. 570
B. 950
C. 1140
D. 5700

Answer

To find the Least Common Multiple (L.C.M.) of 114 and 95, we can use the prime factorization method. Here’s the step-by-step solution:

### Step 1: Prime Factorization of 114
To find the prime factors of 114:
- Divide 114 by 2 (the smallest prime number):
\[ 114 \div 2 = 57 \]
- Next, factor 57. Since 57 is not divisible by 2, we try the next prime number, which is 3:
\[ 57 \div 3 = 19 \]
- Finally, 19 is a prime number itself.

So, the prime factorization of 114 is:
\[ 114 = 2^1 \times 3^1 \times 19^1 \]

### Step 2: Prime Factorization of 95
Now, let's find the prime factors of 95:
- Divide 95 by 5 (the smallest prime number that can divide 95):
\[ 95 \div 5 = 19 \]
- Again, 19 is a prime number.

So, the prime factorization of 95 is:
\[ 95 = 5^1 \times 19^1 \]

### Step 3: Identify the Highest Powers of Each Prime Factor
Now we will list all the prime factors from both numbers and take the highest power for each:
- For 2: The highest power is \(2^1\) (from 114)
- For 3: The highest power is \(3^1\) (from 114)
- For 5: The highest power is \(5^1\) (from 95)
- For 19: The highest power is \(19^1\) (from both)

### Step 4: Calculate the L.C.M.
Now, we multiply these highest powers together to find the L.C.M.:
\[ \text{L.C.M.} = 2^1 \times 3^1 \times 5^1 \times 19^1 \]
Calculating this step-by-step:
- First, calculate \(2 \times 3 = 6\)
- Next, calculate \(6 \times 5 = 30\)
- Finally, calculate \(30 \times 19 = 570\)

Thus, the L.C.M. of 114 and 95 is:
\[ \text{L.C.M.} = 570 \]

### Final Answer
The L.C.M. of 114 and 95 is **570**.

---